As learning solutions reach critical applications in social, industrial, and medical domains, the need to curtail their behavior has become paramount. There is now ample evidence that without explicit tailoring, learning can lead to biased, unsafe, and prejudiced solutions. To tackle these problems, we develop a generalization theory of constrained learning based on the probably approximately correct (PAC) learning framework. In particular, we show that imposing requirements does not make a learning problem harder in the sense that any PAC learnable class is also PAC constrained learnable using a constrained counterpart of the empirical risk minimization (ERM) rule. For typical parametrized models, however, this learner involves solving a constrained non-convex optimization program for which even obtaining a feasible solution is challenging. To overcome this issue, we prove that under mild conditions the empirical dual problem of constrained learning is also a PAC constrained learner that now leads to a practical constrained learning algorithm based solely on solving unconstrained problems. We analyze the generalization properties of this solution and use it to illustrate how constrained learning can address problems in fair and robust classification.

翻译：随着学习解决方案在社会、工业和医疗领域达到关键应用,限制其行为的需要已经变得至关重要。现在有充足的证据表明,没有明确的裁缝,学习可能导致偏向、不安全和偏见的解决方案。为了解决这些问题,我们根据可能大致正确(PAC)的学习框架,发展了限制性学习的普遍理论。特别是,我们表明,强制要求不会使学习问题更加困难,因为任何PAC可以学习的班级都使用经验风险最小化(ERM)规则的有限对应方,也限制学习。然而,对于典型的超美化模型,该学习者涉及解决一个限制的、甚至难以找到可行解决方案的不精密优化方案。为了克服这一问题,我们证明,在温和的条件下,有经验的双重限制学习问题也是PAC制约的学习者,这导致一种实际的限制性学习算法,而现在仅仅基于解决不受约束的问题。我们分析了这一解决方案的概括性特性,并用它来说明限制学习如何在公平和稳健的分类中解决问题。